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Some criteria for nuclearity

Published online by Cambridge University Press:  24 October 2008

M. A. Sofi
M. A. Sofi
Affiliation:
Department of Mathematics, Aligarh Muslim University, Aligarh 202001, India
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Extract

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 100 , Issue 1 , July 1986 , pp. 151 - 159
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1]Crone, L., Fleming, D. J. and Jessup, P.. Diagonal nuclear operators. Studia Math. 43 (1972), 52–56.CrossRefGoogle Scholar
[2]Dubinsky, E. and Ramanujan, M. S.. On λ-nuclearity. Mem. Amer. Math. Soc. 128 (1972).Google Scholar
[3]Dubinsky, E. and Ramanujan, M. S.. Λ(α)-bases and nuclear spaces. J. Indian Math. Soc. 36 (1972), 333–345.Google Scholar
[4]Dubinsky, E.. The Structure of Nuclear Fréchet Spaces. Lecture Notes in Math. vol. 720 (Springer-Verlag, 1979).CrossRefGoogle Scholar
[5]Kamthan, P. and Gupta, M.. Sequence Spaces and Series. Lecture Notes, 65 (Marcel Dekker Inc., 1981).Google Scholar
[6]Kamthan, P. K., Gupta, M. and Sofi, M. A.. λ-bases and their applications. J. Math. Anal. Appl. 88 (1981), 76–99.CrossRefGoogle Scholar
[7]Kamthan, P. K. and Sofi, M. A.. λ-bases and λ-nuclearity. J. Math. Anal. Appl. 99 (1984), 164–188.CrossRefGoogle Scholar
[8]Nelimarkka, E.. On operator ideals and locally convex -spaces with applications to λ-nuclearity. Ann. Acad. Sci. Fenn. A.I. Math. Dissertationes 13 (1977).Google Scholar
[9]Pietsch, A.. Nuclear Locally Convex Spaces (Springer-Verlag, 1972).Google Scholar
[10]Pietsch, A.. Operator Ideals. (North-Holland, 1980).Google Scholar
[11]Ramanujan, M. S. and Rosenberger, B.. On λ(ø P)-nuclearity. Comp. Math. 34(2), (1977), 113–125.Google Scholar
[12]Ramanujan, M. S. and Schock, E.. Nuclearity, bases and finite dimensional decompositions. (Preprint.)Google Scholar
[13]Ramanujan, M. S. and Terzioglu, T.. Diametral dimensions of cartesian products, stability of smooth sequence spaces and applications. J. Reine Angew. Math. 280 (1976), 163–171.Google Scholar
[14]Ruckle, W. H.. Topologies on sequence spaces. Pacific J. Math. 42 (1972), 235–249.CrossRefGoogle Scholar
[15]Ruckle, W. H.Sequence Spaces. Research Notes in Mathematics 49 (Pitman, 1981).Google Scholar
[16]Wojtyński, W.. On conditional bases in non-nuclear Fréchet-spaces. Studia Math. 35 (1970), 77–96.CrossRefGoogle Scholar
[17]Garcia, J. M. L.. ΛN(P)-nuclearity and basis. Math. Nachr. 121 (1985), 7–10.Google Scholar